Filtering in FSK Transmission Systems

At least three stages of filtering are used in the transmission system. A low-pass filter at the transmitter limits the spectrum of the baseband signal (before modulation). In many other modulation methods, the filter may be implemented either at baseband or at RF, after modulation. However, in FSK the nonlinear nature of frequency modulation precludes doing so. The receiver contains a bandpass filter at RF (or intermediate frequency, IF), used to limit the energy applied to the discriminator. After demodulation, a baseband low-pass filter is used to remove as much noise as possible.

In low-cost FSK systems, the filtering philosophy often recognizes that either the transmitter or the receiver is in the subscriber’s home, and this end must be built at minimum cost. Placing most of the filtering at the headend helps keep the cost of the home equipment down. This somewhat compromises the noise performance or bandwidth efficiency in favor of low cost. For example, a data communications system that communicates from the headend to addressable set-top converters will employ significant filtering at the transmitter. At the receiver, the IF filter frequently employs a low-cost ceramic filter made for FM receiver use (180- to 200-kHz bandwidth), and the deviation is set to make the occupied spectrum match the width of the filter.

Manchester Encoding

We now describe one method for adding a clock to the data signal and for removing any dc component. The method is independent of the modulation but is introduced at this point because it is often used with FSK modulation. Manchester encoding is also used in 10/100 BaseT Ethernet. In data transmission, it is necessary not only to recover the transmitted data but also to recover a clock signal, used to define when a data state should be sampled. A clock is necessary in order to allow the receiver to “know” when a new bit arrives. The clock must be recovered from the transmitted data, which can be done in many ways. Manchester encoding is often used in low-cost systems due to its simplicity. The data is said to be “self-clocking.” The penalty is that the spectral efficiency of the transmission is halved compared with what it would be otherwise.

Figure 4.2 illustrates one form of Manchester encoding, which is applied to the data before it enters the modulator. At the top of the figure is a string of data represented in NRZ (nonreturn to zero) format, in which the data bits appear one after another, with nothing in between. A separate clock signal is typically used to transfer NRZ data from one place to another. However, in a communications link, a separate clock path does not exist. Therefore, in some manner the clock must be embedded with the data. Manchester encoding provides a simple, albeit bandwidth inefficient, way to do this.

Figure 4.2 Manchester Encoded Data.

The center portion of Figure 4.2 shows the bit cell boundaries: the time when the signal transitions from one bit to the next. At the bottom of the figure is shown the Manchester encoded data. In the middle of each bit cell, Manchester encoding causes the signal to transition from one level to the other. This regularly spaced set of transitions forms a clock for the receiver. The first half of each cell contains the value of the bit; the second half contains the complement (opposite state) of the bit (or vice versa). A variation is differential Manchester encoding: At the beginning of each cell containing a 0, the encoded signal makes a transition; at the beginning of each cell containing a 1, the transition is omitted. There is always a transition in the middle of the bit cell. On the other hand, if the transition occurs in the middle of a cell containing a 1 but not in a cell containing a 0, the encoding is called biphase mark. It is used in the S/P DIF interface described in Chapter 22.

Note that Manchester encoding is not a modulation method. Rather, it is a baseband coding method that is useful in low-cost systems to provide clock recovery and removal of a dc component of the data.

Basic Math of BPSK

One might be tempted to look at the waveform of Figure 4.3 and conclude that BPSK is a phase modulation technique because the carrier reverses phase between transmission of 1s and 0s. Indeed, the name of the modulation leads to that conclusion. The correct way to visualize BPSK is as double sideband suppressed carrier amplitude modulation.

To understand the modulation process, assume that the data signal is an alternating series of 1s and 0s that has been filtered to a sine wave, following the logic of earlier sections of this chapter. Then the modulating signal can be expressed as

(4.2)

where the subscript m indicates the modulating frequency and ω_m = 2πf_m. As implied by Figure 4.3, the modulation process involves multiplying this modulating frequency by the carrier frequency, represented as

(4.3)

The resulting modulated signal is expressed as follows, expanded through use of common trigonometric relations:

(4.4)

The center row shows the phase shift with modulation: When the sinusoid representing the data modulation (sin ω_mt) is positive, then the carrier has one phase; when it is negative, the carrier phase is reversed, as illustrated in Figure 4.3.

The bottom expression shows that the frequency components of the modulated signal exist at the carrier frequency ω_c minus the modulating frequency ω_m and at the sum frequency. Thus, the bandwidth occupied by the signal is twice the modulating frequency, as is the case with conventional double sideband amplitude modulation. A component at the carrier frequency does not exist.

The mathematics of the synchronous detector shown in Figure 4.3 consists of multiplying Equation (4.3) by the carrier sinusoid and applying more common trigonometric relationships:

(4.5)

The first term in the bottom line represents the recovered modulation; the second term represents components surrounding twice the carrier frequency. This latter term is filtered in LPF2 of Figure 4.3.

The Constellation Diagram

A particularly useful representation of QPSK and higher-order signals is the constellation diagram, shown at the bottom of Figure 4.4. It is formed by routing the demodulated I signal, before slicing, to the x axis of an oscilloscope and the Q signal to the y axis. Each of the four possible positions of the symbol is formed from the I- and Q-channel data, with the position along the x axis being determined by the I data and that along the y axis by the Q data. The amplitude of the signal is proportional to the length of the vector from the origin to any one of the four states, which are equidistant from the origin. Note that, in the absence of any transmission impairments, the four points in the constellation are small points. They are enlarged in the illustration to make them easier to see. As the signal transitions from one point in the constellation to another, the trajectory may pass through several other points.

The y axis forms the decision threshold for the I channel since it is halfway between the I channel states. Similarly, the decision threshold for the Q channel is defined by the x axis. The decision threshold defines the signal state at which the comparator(s) for that axis makes the decision that the bit represents one value or the other. If the point in the constellation crosses the decision threshold due to noise or other channel impairment, the comparator will yield the wrong output, resulting in an error in transmission. Computation of bit error rate (BER) is based on determining the probability that any one point in the constellation could be forced to cross a threshold into another state’s place in the constellation. When this happens, at least one, and sometimes more, bits are received in error. Obviously, the closer together the states in the constellation (i.e., the “denser” the modulation), the less noise or other disturbance can be tolerated before a state can be misinterpreted as that of a neighbor.

The peak amplitude of the signal is equal to the length of a vector from the origin to the farthest point in the constellation. In the case of QPSK, all four points are equidistant from the origin, so any of them can be taken as the peak amplitude. Note that the constellation diagram represents voltage, not power.

Though the QPSK signal has four states at the same amplitude, between states the amplitude of the carrier drops, due to filtering at the modulator. This filtering is represented by LPF1 and LPF2 of Figure 4.4. Also see Figure 4.3 for an illustration of the resultant modulated waveform. Since QPSK is nothing more than two BPSK signals combined 90° out of phase, then the same wave-form of Figure 4.3 would generally apply to a QPSK signal: Between symbols, the signal amplitude passes through 0 if the state of both bits in the symbol changes. This usually doesn’t cause much concern in cable television work. However, in the case of systems having high AM/PM conversion, where a change in amplitude of the signal can cause a change in its phase, the amplitude of the signal dropping to 0 can be a problem. AM/PM (spoken as “AM to PM”) conversion is a phenomenon in which changes in the amplitude of a signal result in a change to the phase of the signal. In data communications, it can distort the phase reference for the PLL in a receiver, making accurate recovery of the data less likely. AM/PM conversion can be caused by asymmetrical nonflatness of the channel and by capacitance modulation in an amplifier that is operating out of its linear range.

A satellite communications link is susceptible to AM/PM conversion. Often the power amplifier on the satellite is driven slightly into saturation and may exhibit AM/PM conversion. To reduce the effect, satellite communications often uses offset QPSK, in which the transition times of the data in the I and Q channels is offset by one-half of a bit cell. Since the two channels don’t change amplitude at the same time, the amplitude tends to be more constant.

Binary Numbering

Before we discuss higher-order QAM, we need to review a couple of elementary concepts in binary numbering. We will show groups of two bits whose value is designated as 00, 01, 10, or 11. The correct way to speak this is “zero-zero,” “zero-one,” and so on (“zero” may be replaced with “oh”). These numbers are not to be confused with the base-10 numbers that look the same. If confusion between base 2 (binary) and base 10 (our familiar numbering system) is possible, then the number will be followed by a subscript 2 or 10. The following are equivalent numbers in base 2 and base 10: 010₂ = 2₁₀ and 1101₂ = 13₁₀.

16-QAM Modulation

The next denser modulation format commonly found in cable television work is 16-QAM (16 state quadrature amplitude modulation). It is used for modem return services that require higher bandwidth efficiency than that offered by QPSK. It is similar to QPSK except that each axis is divided into four equidistant levels so that each channel (I and Q) can carry the information of two bits. Each of two phases carries two bits at a time, so there are four bits per symbol.

Figure 4.5 illustrates a basic 16-QAM modulator and demodulator. Note the similarities between this and the QPSK modulator of Figure 4.4. Each channel of the 16-QAM modulator can take on not only the two phases of QPSK but, in addition, intermediate amplitude values. Data is split into two channels, the I and Q channels. Instead of only one bit being routed to each channel at a time, two bits are routed to each. They are added as shown in the table beside the Q-channel modulator: for example, if the two bits are 11 (pronounced “one-one,” since we are working in binary numbers), then the output of the adder is +3, and the Q-channel modulator puts out an amplitude of +3, with a positive phase (calling this the “positive” phase is arbitrary). If the data set is 10 (one-zero), then the adder supplies an analog signal of amplitude +1 to the modulator.

Figure 4.5 16-QAM Modulator and Demodulator.

Similarly, a 01 data set yields an analog level of −1, which causes the modulator to supply a 1-level signal with the opposite phase. A 00 data set will cause the adder to supply −3 to the modulator, producing a higher-amplitude negative phase signal (saying the phase is “negative” is an arbitrary way to refer to the fact that the phase is opposite that of a “positive” phase). By convention, the inner levels are +1 and −1. To retain equidistance, spacing between levels is 2.

The same thing takes place in the I channel. Each channel can thus take on any of four states. With the I channel in any of four states and the IQ channel independently in any of four states, a total of 16 states are possible. This is the number of states required to represent 4 bits (2⁴ = 16). The modulation technique, 16-QAM, derives its name from the 16 possible states of the modulated signal.

The receiver contains in-phase and quadrature demodulators, as does the QPSK receiver. However, the output of each demodulator must be further processed to recover two bits. This is done for the I channel, using the three data slicers shown. The data slicers have thresholds set by the four voltage divider resistors, at levels of T1, T2, and T3. These levels are set halfway between the received states in the I channel. The thresholds are shown in the constellation diagram at the bottom of Figure 4.5. By reading the outputs of the three comparators, A1 through A3, the decoder can deduce the transmitted state. Note that there are more decision thresholds than there were in QPSK, because there are more points in the constellation, and we must have a way to distinguish between all states. We often refer to decision boxes rather than thresholds, because each point in the constellation has its own box it must be in, defined by the I and Q decision thresholds.

64-and 256-QAM

For downstream transmission, the industry employs modulation formats that are similar to 16-QAM except that they have more states. They transmit more bits per symbol, so the constellation is denser. The distance between states is smaller, so the susceptibility to noise and transmission path distortion is greater. On the other hand, the spectral efficiency is greater, meaning that more bits can be transmitted in the same RF bandwidth.

Figure 4.6 illustrates the constellation diagrams for 4-QAM (i.e., QPSK), 16-QAM, 64-QAM, and 256-QAM. The diagrams are drawn to the same peak scale, so it is easy to see the higher density of the higher-order modulation formats. Also shown for comparison are several characteristics of the different modulation formats. Most of the characteristics are shown measured in decibels with respect to a QPSK signal.

Figure 4.6 Constellation Diagrams, 4-, 16-, 64- and 256-QAM.

For each modulation type, the approximate power peak-to-average ratio is shown first. This is computed by averaging the power of each state, ignoring the effects of filtering. Later we show how this is done. A more precise peak-to-average ratio must include the effects of filtering. Next is shown the theoretical spectral efficiency, in bits per hertz. The spectral efficiency is a figure of merit, defining how many bits per second can be transmitted per hertz of RF bandwidth. The actual spectral efficiency will be less than this, due to the practical need for excess filter bandwidth.

Below the peak-to-average ratio is shown the relative distance between a state and the closest state boundary, expressed in decibels. The relative distance between a state and the boundary is defined in the QPSK constellation, as is the peak amplitude. It is the decibel measurement of how much closer to each other the states are in the denser modulation formats. The reference is a QPSK constellation, and the distance is normalized to the peak, not the average, amplitude.

The relative amplitude entry is derived from the BER vs. signal-to-noise ratio of the equation graphed in upcoming Figure 4.17. It is an indication of the increase in signal-to-noise ratio required to maintain a 10⁻⁶ bit error rate. Recall that the signal level to which we refer here is the average, not the peak level.

Figure 4.17 Bit Error Ratio vs. E_b/N₀ for Several Levels of QAM.

32-QAM Modulation

A similar modulation format to those shown earlier is 32-QAM, which has 32 states and a theoretical spectral efficiency of 5 bits per hertz. Two different 32-QAM constellations are shown in Figure 4.7. Figure 4.7(a) is distinguished from the 64-QAM formats shown earlier in that every second state is occupied. This puts more distance between states, resulting in easier decoding. Figure 4.7(b) illustrates an alternate encoding of 32-QAM, where the four corner states are omitted. Both encoding methodologies have been reported in the literature.

Figure 4.7 32-QAM Constellations. (a) Alternate states. (b) Corners removed.

A total of 32 states represents 5 bits (2⁵ = 32). With 16-, 64-, and 256-QAM, each axis, I or Q, can represent a certain number of bits, because the number of states along each axis is a power of 2. On the other hand, 32-QAM does not have this correspondence between bits and states possible. The five bits making up a symbol can, however, be encoded into one of the 32 possible states.

Bit Rate and Baud Rate

Perhaps the most common mistake made in communications terminology today is to confuse the rate at which data is transmitted with the rate at which symbols are transmitted. The rate at which bits are transmitted (the bit rate) is measured in bits per second. In most modulation formats, however, more than one bit is transmitted at a time. The bits that are transmitted simultaneously are grouped into a symbol. The rate at which symbols are transmitted is the baud rate.²

In DOCSIS upstream transmission, the lowest specified symbol rate is 160 kbaud. If QPSK modulation is used, the bit rate is twice that, or 320 kb/s, because there are two bits transmitted per symbol. However, if 16-QAM is used, the bit rate is four times the baud rate, or 640 kb/s. (Note that the plural of baud is baud, without an “s.”) As shown earlier, 256-QAM modulation groups 8 bits into a symbol, so the baud rate is one-eighth of the bit rate. The occupied bandwidth of the signal is a function of the baud, or symbol, rate, because the baud rate tells us how many times the modulation state changes each second.

Filtering Methodology

This section introduces the filtering methodology that is followed in many data communications applications and for many types of digital modulation (except FSK, which was covered earlier). The filters at the transmitter and receiver are effectively in cascade in the signal path; together it is desirable that they form a filter just wide enough to convert the highest modulating frequency into a sine wave.(As a practical matter, the filter will be somewhat wider than this.) A particularly useful class of filters is called raised cosine filters by the data communications industry. They are called sine squared filters by the television engineering community.

The equivalence is seen in the following trigonometric relationship:

(4.6)

The left side of the equation shows the rationale behind the television usage of sine squared terminology, while the right side shows the basis of the data communications term raised cosine. The 1 on the right “raises” the cosine term, which by itself ranges from −1 to +1. In television work, a sine squared filter is used to produce the well-known sine squared pulse and bar, which are useful in analyzing analog video channel performance. In data communications, the cascaded performance of the low-pass filters at the transmitter and receiver, possibly in addition to the response of the IF filters, form a raised cosine filter, which ensures a fast transition between the modulation states, consistent with minimal spectrum occupancy.

It is desirable to heavily filter at the transmit end in order to reduce the spectrum usage, and heavy filtering is desirable at the receive end to minimize noise added during transmission. However, the total filtering applied must conform to the raised cosine shape. In many communications systems, the designer places one-half of the filter function at each end of the communication channel. When this is done, the filter response at each end is the square root of the complete filtering function, so the filter used on each end is often described as a root raised cosine filter. Note that the filters at the transmit and receive ends must be matched to each other and collectively are often referred to as matched filters.

Note that any bandpass filter must be symmetrical with respect to the center (carrier) frequency for quadrature signals (that is, any QAM signal). This is a requirement of quadrature transmission systems. If the overall filter response is not symmetrical, we can show that the carrier phase will be recovered incorrectly, resulting in a loss of signal integrity. In addition, the in-phase and quadrature channels will crosstalk.

Required Bandwidth of the Signal

We shall develop the concept of occupied bandwidth using a BPSK signal, and then show how it is extended to any QAM modulation format. The bandwidth requirements of BPSK can be understood by considering the worst case, that of an alternating string of 1s and 0s. This will produce a square wave, one cycle of which is composed of an alternate 1 and 0, or two bits, as explained earlier. Thus, the frequency of the square wave, f_m, is one-half the bit rate. Figure 4.10 illustrates the spectrum of the data signal at baseband. The square wave of alternating 1s and 0s illustrates the highest frequency of the data signal. From Fourier analysis, we know that a square wave consists of all odd harmonics of the fundamental frequency, as shown by the vertical lines in the spectrum plot of the figure.

Figure 4.10 Baseband Spectrum of BPSK Signal.

In real data systems, the data is not an alternating series of 1s and 0s. Rather, it is all possible combinations of strings of 1s and 0s. The resulting spectrum will fill in the spectrum from the maximum frequency, f_m, down to close to 0 frequency (and all harmonics thereof).

However, the harmonics are not needed to convey the information contained in the datastream. If all frequencies above f_m are removed by filtering, then an alternating series of 1s and 0s will produce a sine wave of frequency f_m. Transitions that occurred at a slower rate (i.e., from a series of like states) would follow a sinusoidal transition path but would be flattened on top.

The theoretical (not practical) spectral efficiency of a BPSK signal is developed as follows. The alternating 1-0 sequence defines the highest frequency of the data baseband signal. If it is filtered by removing all harmonics, leaving a sine wave, then the frequency of the sinusoid is one-half the data rate (one cycle is composed of a set of a single 1 followed by a single 0). The bandwidth of the modulated signal is twice the modulating frequency (the sum less the difference between the carrier and the modulating frequency), which in turn is equal to one-half the bit rate. Thus, the spectral efficiency is two times one-half the data rate, or one times the bit rate, one bit per hertz.

A filter that completely cuts off just above f_m may be ideal but is hardly practical. A practical filter has the raised cosine shape and is somewhat wider than f_m. The sharper the filter cutoff (that is, the less bandwidth required between the filter passband and the stopband), the less bandwidth is required. However, the sharper filter is more difficult to build, and more group delay (see Glossary) is likely, depending on the realization. The shape of a real filter is described by the alpha of the filter, where alpha is defined as the excess of occupied bandwidth compared with the minimum required bandwidth:

(4.7)

Recall that the common filtering practice is to put the square root of the raised cosine filter response at the transmitter, and the same at the receiver. For example, as shown in Chapter 5, the filter specification for the transmitter of a DOCSIS-compliant downstream modulator operating in the 64-QAM mode is an 18% root raised cosine filter, meaning that the filter is 18% wider than the minimum required bandwidth.³ This may also be interpreted that the occupied bandwidth is 18% wider than the theoretical minimum.

QPSK works the same way as does BPSK as far as the spectrum is concerned. QPSK is simply two BPSK carriers at the same frequency, but 90° out of phase with each other. The spectra overlap; and since a spectrum analyzer doesn’t respond to the phase of a signal, the BPSK and QPSK spectra are very hard to tell apart.

Higher orders of QAM modulation are simply the same thing as QPSK, with the exception that rather than each of the two quadrature carriers taking on only two states (phases), they can take on multiple amplitude and phase states. The spectrum is again very difficult to distinguish from that of a BPSK signal. The bandwidth is based on the number of symbols transmitted per second, not on the number of bits transmitted per second. See the earlier discussion of bit rate vs. baud rate.

Clock Recovery

While self-clocking formats, such as Manchester encoding (see Section 4.2.2), could be used with BPSK and higher-order modulation, this is normally not done. The higher number of transitions associated with Manchester encoding would double the required bandwidth of the signal. Rather, an NRZ datastream is transmitted. Clock recovery consists of synchronizing a clock to the transitions in the received data. Where no transition occurs, the clock “freewheels” through the place where the transition should occur. If a long transmission of 0s or 1s occurs, it is possible to lose clock synchronization, so “scrambling” or some other technique is often used to force a state change every so often.

Data Scrambling

Scrambling is often used as a method of dc removal, which is necessary to effect proper operation of a number of circuits in the receiver, including clock recovery. Scrambling in its data transmission sense is not the same as when used in the cable television sense. In cable television, the word refers to the process of rendering a video and/or audio signal unintelligible to a normal receiver. A special circuit must be used to restore the signal to its intelligible form.

In data transmission, scrambling is the process of combining a datastream with a pseudorandom bit sequence, with the intent to remove any possible dc component of the datastream. An equivalent way of expressing the function of the scrambler is to say that it equalizes the power density over the occupied bandwidth. It can be shown that, if any data signal is exclusive ORed with a pseudorandom signal having a flat spectrum and no dc component, then the resultant signal will have a flat spectrum and no dc component. The signal can be recovered at the receiver by exclusive ORing the signal with the same pseudorandom signal.

The exclusive OR operation is a logical operation on a single bit of data. If the two bits are the same, the output of the exclusive OR operation is a logical 0; if the two bits are different, the output is a logical 1.

Differential Encoding

The phase locked loop used for carrier recovery in Figure 4.3 exhibits a problem, in that a phase error ambiguity exists: If the output of LO2 divided by 2 is in phase with the carrier applied to M1, then the recovered data will have one polarity; if it is out of phase, the recovered data will have the opposite polarity. Since the loop is closed at twice the frequency of the carrier, no simple way to determine the correct phase exists. One way would be to detect a code word buried in the data and to correct for a phase error.

A method frequently used to resolve the phase ambiguity is differential encoding. In differential BPSK, a 0 is represented by no change in phase from the previous state; a 1 is represented as a 180° phase change. The detector is arranged to compare the phase of the previous bit with the phase of the present bit; if the phase doesn’t change, a 0 is delivered as the received bit. If the phase is changed, a 1 is delivered (or vice versa). With more complex modulation schemes, sometimes a recognizable binary word is used to determine the correct phase.

The Eye Diagram

The recovered waveform at the “monitoring” point shown in Figure 4.3 should be the same as the waveform applied to modulator M1. However, distortion in the transmission path will change the wave shape, as suggested in Figure 4.11 (the dotted line is the original wave shape; the solid line is the received wave shape). Noise will put “ripple” on the waveform, as seen on an oscilloscope. If enough noise is added, then the state of the signal may be misinterpreted, producing a bit error. Besides noise, group delay can distort the wave shape, as can echoes and amplitude tilt in the channel.

Figure 4.11 BPSK Received Waveform and Eye Diagram.

A particularly useful oscilloscope display is called an eye diagram, so named for its resemblance to an eye. It is produced by supplying the received waveform (before the slicer) to the vertical input of the oscilloscope. The horizontal time base is set to display more than one bit cell and is triggered by the recovered clock.(For simplicity, the diagram shows only one bit cell width, though typically some additional time would be displayed.) When a bit is undergoing a transition from 0 to 1 at the trigger time, the waveform will be positive-going on the left. When it is in transition from 1 to 0, it is negative-going. It may also be the same as the last bit, in which case it forms a flat line at the top or bottom of the display. Similarly, the display toward the right is formed by transitions to the next bit after one displayed.

The slicer interprets the signal to be either a 1 or a 0. The sample time should be in the center of the bit cell, when the eye is as “open” as possible. One of the practical error sources, which is often not considered in computing BER versus noise, is errors in the timing of the sample: If the timing is in error because of problems in the clock recovery circuit (perhaps due to noise, interference, or waveform distortion), then it is more probable that the bit will be misinterpreted. A figure of merit is the eye opening, or the percentage of the peak (fully opened) eye that is seen as open in the eye diagram.

Eye closure can be caused by a number of transmission problems. The effect of noise is illustrated in Figure 4.11. Besides noise, errors in the filter shape can cause pulse distortion, which will tend to close the eye. If the transmission path exhibits excessive group delay or if an echo exists, then one bit may not completely “go away” before the next bit starts. This condition is called intersymbol interference, or ISI.

Illustrated in Figure 4.11 is a potential bit error, in which a noise spike carries the waveform so far from its correct position that it actually crosses the decision threshold of the slicer. If the clock causes the slicer output to be read at the time of the noise spike, then a bit error would occur.

We have illustrated the eye diagram for a BPSK signal. When quadrature modulation is used, such as in 4-, 16-, 64-, or 256-QAM modulation, the eye diagram is displayed separately for each axis of modulation. When there are more than two states per symbol, then the eye diagram will be more complex but will nonetheless resemble that of Figure 4.11.

Effects of Interference on the Constellation Diagram

The constellation diagram is useful for understanding the effect of interference on the signal. In Figure 4.12, quadrant I illustrates the effect of a continuous wave (CW) interfering signal in the passband of the receiver. An interfering carrier creates a beat note, which forces each point in the constellation to spread out in a circle around the desired point. The radius of the circle is proportional to the relative amplitude of the interfering carrier. Unlike analog video signals, the impact of the interfering carrier tends to be independent of its location within the passband, except for some effect caused by nonflatness of the receive filter.

It is worth noting that composite second order (CSO) and composite triple beat (CTB) are not CW interferences. Their amplitudes change constantly, with peaks reaching 15 dB above the average. The variation is due to phase alignment of the multiplicity of carriers. The occurrence of a peak is rare, but the peak can last from a microsecond to a few hundreds of microseconds (longer than the burst protection period of the Interleave.⁴

Quadrant II illustrates the effect of phase noise on the signal. Phase noise is introduced when a local oscillator has significant phase noise. It can also be shown to occur when the transmission channel does not have a response that is symmetrical about the carrier. Phase noise causes the constellation to rotate about the origin.

Random noise (quadrant III) causes each point in the constellation to “blur,” somewhat as with CW interference, except that the spreading of the point tends to be more nearly uniform within the circle, and the circle does not have a well-defined radius. Again, recall that the spreading of points in the constellation becomes a problem when the spread points cross a threshold to another state. With QPSK, that threshold is defined by the axes of the constellation diagram. With more dense modulation, the thresholds are closer together, so lower levels of interference will cause the threshold to be crossed. As a practical matter, an error may occur when a point is forced close to a threshold boundary: Real demodulators don’t work quite as well as the theory predicts.

Finally, reflections, which cause ghosting in analog transmission, cause a replication of the constellation around each point in the constellation, as illustrated in quadrant IV. The constellation is rotated with respect to the “direct path” constellation, due to phase shift between the direct and reflected RF carriers. Usually, systems that employ denser modulation formats include adaptive equalizers, which can compensate for echoes if the echo doesn’t change too quickly.

If the peak amplitude of the signal is 1, as shown in quadrant IV, then the distance from the nominal location of any of the four states to the nearest state boundary is 0.707. This result is easily obtained via simple geometry, where the amplitude vector, of length 1, is the hypotenuse of a right triangle and the distance to the nearest decision threshold is one of the two other sides of the triangle. Bear in mind that it is more common to measure the amplitude of a digital signal by using the average amplitude, which concept is explored later.

Effect of Gain Compression on the Constellation Diagram

Gain compression can occur in any amplifier stage, but it is most likely to be recognized as such if the amplifier is handling only a single digitally modulated signal. Gain compression may be described as a voltage transfer function having second- and third-order components, just as when we are describing distortion in broadband amplifiers. As the input signal instantaneous amplitude increases, the output amplitude increases more slowly. If an amplifier is handling multiple signals, this phenomenon produces composite second- and third-order distortion.

Figure 4.13 illustrates the effect of gain compression on the digitally modulated signal. The uncompressed constellation points are represented as dots, and the compressed constellation points are represented as pluses. In both cases we have normalized the signal amplitude to the same average power. This represents what a receiver would do if its automatic gain control (AGC) responded to average power. The decision thresholds are at the same point for both the uncompressed and the compressed constellations, due to the normalization. Note that the uncompressed constellation points are in the center of the decision boxes, as far from the decision boundaries as possible, where they should be for best data recovery.

Figure 4.13 Effect of gain compression on a 64-QAM signal.

The compressed constellation has very many points near the edge of the decision boundary; a slight amount of noise or other impairment would drive the constellation point outside of the boundary, forcing a bad decision on the transmitted information. For illustration, we have chosen a rather large amount of gain compression. But you can see that the problem gain compression causes is that the constellation is moved from the optimum location, with the result that more errors will be made for a given impairment level.

Peak-to-Average Ratio

Higher-order modulation techniques exhibit a higher peak level than the average level read on a power meter. Figure 4.14 illustrates one reason for this, based on a 16-QAM signal. It shows the same constellation diagram as depicted in Figure 4.5. The constellation is scaled such that the amplitude of any one of the four corner states is 1. This represents the peak level of the signal. The chart in the figure computes the average signal level based on equal occupancy of any of the 16 possible states. Since all four quadrants behave the same way, it is necessary to average over only one quadrant.

Figure 4.14 Peak and Average Level of a 16-QAM Signal.

The voltage amplitude of each state is determined by simple geometry, assuming that the highest-amplitude state has an amplitude of 1. From the geometry of a right triangle, it then follows then that the coordinates of that state are (0.707, 0.707). The state closest to the center has a distance on each axis of one-third of this, so its total distance is one-third that of the farthest state. States 2 and 3 each have one coordinate of 0.707 and one coordinate that is one-third of that. Application of the Pythagorean theorem shows that the magnitude is

When looked at on a voltage basis, the voltage average is the average length of the four vectors that define the states in the quadrant. The power average is obtained by squaring the voltage vector for each state, because power is equal to the square of the voltage divided by the resistance:

The resistance would cancel out in the computation, so it is omitted. The right-hand column in the chart in Figure 4.14 computes the power peak-to-average ratio, based on squaring the voltage vectors.

Effects of Fitering on Peak-to-Average Ratio

The peak-to-average ratio is not completely characterized by the state distribution as implied, however. It is also a function of the filtering at the transmitter. This was illustrated in Figure 4.3, which shows an amplitude dip in a BPSK signal when the transmitted bit changes states. The same thing happens with higher-order modulation techniques, and that somewhat changes the peak-to-average ratio. The peak-to-average ratio is of concern in cable systems, particularly in the operation of return plant, where a limited number of digital signals comprises the return spectrum handled by the return laser.

It was shown that the peak-to-average ratio is partially a function of the relative amplitude of each state and the percentage of time the signal spends in each of those states. Another factor, not yet taken into account, is the filtering applied to the signal. As the filtering becomes more aggressive (i.e., lower alpha, corresponding to smaller excess RF bandwidth), the filter tends to ring more, meaning that the signal will be of even greater amplitude than predicted, for a small percentage of the time.

Ito⁵ studied this theoretically for a filter having an alpha of 11.52%. He computed the percentage of time the power of a 64-QAM signal exceeded its average by various amounts. The curve follows the percentage computed based on state occupancy, until the percentage of time is less than the percentage of time the signal spends in the highest state. He found that the signal exceeded the average by 6 dB or more for about 0.1% of the time and was 9 dB above the average for about 0.00001% of the time. (For reference, 6.25% of the time a 64-QAM signal is in its highest-amplitude state, which is 3.68 dB higher than the average power.)

Multiple-carrier systems, such as OFDM (described in Section 4.3.3), had a much higher peak-to-average ratio. It exceeded the average by 12 dB for 0.00001% of the time. This is because multiple-carrier modulation signals tend to have the amplitude distribution of random noise, which has a finite probability of being very high for a very short time. The subject of peak-to-average ratio is revisited in Chapter 16.

Effect of Linear Distortion on Pulse Shape

Linear distortions that can enter the cable plant include echoes, also called reflections, in which a pulse is reflected from an impedance mismatch and adds constructively or destructively to the original waveform. Another linear distortion is group delay, in which the delay of the channel is not flat with frequency. Yet another linear distortion is amplitude ripple, in which the frequency response of the channel is not flat. All of these linear distortions are familiar to those who have dealt with them in analog modulation. In digital modulation, these distortions tend to close the eye of the data (illustrated in Figure 4.11) and can present a serious impediment to data recovery.

Figure 4.15 illustrates pulse distortion caused by a single reflection. In the modeling, the reflection was delayed about 75% of the pulse duration. In many cases the pulse would be delayed much more. Also, to make it easier to see the effect of the delayed pulse, we have assumed a much wider excess bandwidth than is normal. You can see that the pulse is significantly distorted, which would result in significant closure of the eye diagram. Since the delay in the reflected signal may be much more than one symbol time, we often refer to these distortions as producing intersymbol interference, or ISI.

Figure 4.15 Pulse distortion caused by a reflection. (a) Original pulse. (b) Distorted pulse.

Adaptive Equalization

Because ISI can be so serious a problem in dense modulation formats, it is desirable to compensate for it at the receiver. This is done with an adaptive equalizer, a circuit that produces an echo that is the same as the echo added by the system, except the intentionally produced echo is subtracted from the signal.

The higher the density of the modulation (e.g., 256-QAM), the more important the adaptive equalizer is, because less eye closure can be tolerated before bit errors occur.

Figure 4.16 illustrates an adaptive equalizer installed in each channel of a QAM demodulator. Figure 4.16(a) shows a QAM demodulator, with an adaptive equalizer installed in each channel, after the demodulator and before the data slicers. Thus, we have an analog signal at the point where the adaptive equalizers work.(Note that in modern circuitry, the signal is often digitized before detection, and the detection, filtering, and equalization are performed using digital signal-processing techniques. The point is that the signal at the adaptive equalizer is a continuous signal, even if it is represented digitally. It has not been reduced to the final binary value, which is done in the data slicers and the encoder.)

Figure 4.16 (a) Location of adaptive equalizer in receiver. (b) Adaptive equalizer.

Figure 4.16(b) shows a 24-stage adaptive equalizer. The signal from the demodulator enters at the left and is successively delayed by small intervals (usually one symbol in time), designated Z⁻¹. After each delay, the signal amplitude is adjusted by complex gains A1, A2, etc. The signal with gain adjusted is added to the main signal, which is shown coming through A1 before any delay, plus the output of all other stages of the adaptive equalizer. Note that the gains A may be negative values, to subtract signal if necessary; depending on the RF phase of an echo, the reflected signal may add, subtract, or have any phase angle in between.

Thus, the adaptive equalizer generates an echo similar to that introduced in the plant and then subtracts it from the main signal in order to cancel the echo. The small eye diagrams at the left and right illustrate the effect of the adaptive equalizer (for simplicity, we are showing a BPSK eye, but the adaptive equalizer is more likely to be needed for dense modulation). Of course, it is necessary to adjust the value of each “tap” (each gain A) to compensate for whatever deficiency exists. The gains A may be positive or negative. It is the job of the detector, shown by each adaptive equalizer, to determine the eye closure and to adjust the taps until the eye diagram is optimized.

Adaptive equalizers are useful for compensating for echo and also for compensating for group delay and amplitude response. In some instances the main tap will not be the first one, because preechoes are possible. Amplitude errors in the channel may be modeled as a pair of echoes, one before and one after the main signal, so the detector may have to assign another tap the job of passing the highest signal (highest value of A), in order to have some preecho taps to use in canceling amplitude errors. A common combination is 7 preecho, the main tap at the eighth position, and 24 posttaps.

In the past it has been most common to place the adaptive equalizer in the receiver, as is done in digital video receivers. However, DOCSIS modems have a unique need in the upstream direction that has caused the equalizer to be placed in the upstream transmitter (at the modem) and the detector in the Cable modem termination system (CMTS). Since signals from each modem travel a partially different path, they are subject to different distortions, depending on that path. Thus, each modem will require a different adaptive equalizer setting. A detector in the CMTS monitors the eye opening for each modem and sends signals to that modem telling it to adjust its adaptive equalizer filter until the eye opening is maximized. After that has been done, the eye opening is checked on each transmission, and changes are commanded from the CMTS to the modem when necessary.

The Concept of E_bN₀

For a given modulation format, as one increases the data rate, the required transmission bandwidth will increase proportionally. If a 1-Mb/s datastream requires 0.5 MHz of transmission bandwidth, then a 2-Mb/s datastream will require 1 MHz of bandwidth. The bandwidth required has doubled, doubling the noise that of necessity must be admitted by the filtering, but the data rate has also doubled.

In order to compare the performance of different modulation formats, which may be used to transmit different bit rates, we use the concept of energy per bit divided by noise power per hertz, E_b/N₀. This parameter, pronounced “E B over N zero,” removes data rate and noise bandwidth from the expression. It serves the same purpose as a carrier-to-noise ratio in analog video transmission, but it is not the same as carrier-to-noise ratio. One can think of this parameter as the average carrier-to-noise ratio per bit. In a given case, one can convert to carrier-to-noise ratio by multiplying E_b/N₀ by the bit rate and then dividing by the noise bandwidth. In logarithms, this is done as follows:

The use of E_b/N₀ permits one to compare the performance of different modulation systems without having to correct for different bit rates between the systems.

BER Versus E_bN₀

The bit error rate of various levels of QAM modulation can be approximated by the equation⁷

where

The error function, erfc, is a mathematical expression of the probability that a normal random variable (such as random noise) will exceed a certain level.

Graphing this expression for a number of levels of QAM yields the chart of Figure 4.17. This graph is normalized for a carrier-to-noise ratio, expressed as E_b/N₀, the energy per bit divided by the noise per hertz of bandwidth. This is a common expression for carrier-to-noise ratio, which is independent of bit rate and bandwidth. This graph does not include the effects of other impairments from the signal source, the transmission channel, or the receiver.

Note that digital communications engineers often refer to this as signal-to-noise ratio rather than carrier-to-noise ratio. Cable television engineers often reserve the term signal-to-noise ratio, for the baseband measurement and use carrier-to-noise ratio for the RF measurement. Digital communications engineers don’t always make this distinction.

Bit Error Rate

BER is simply the average number of bits received in error divided by the total number of bits received. Scientific notation is used to express BER, since the number of errors had better be quite small. If one bit out of every million received is in error, then the BER is

Note that BER is dimensionless. In order to measure BER in a laboratory setting, usually a known bit pattern is transmitted repeatedly. The receiver compares the received bit pattern against what it knows was transmitted and then displays the BER. It can take a long time to measure BER when the BER is low. Suppose it is desired to measure the BER of a 64-QAM signal being transmitted at a rate of 30.342 Mb/s. If the BER is 10⁻¹⁰, then we will wait, on the average, through 10¹⁰ bits before we receive a bad one. This will require

before one error is received. We must receive a number of errors before we can conclude that we have a statistically valid sample. You can see that it will take a long time to make this measurement.

In-service BER cannot be performed on a known bit pattern because, by definition, the receiver cannot know what the transmitter is sending, unless the transmitter and receiver are located at the same place and the receiver is also getting the bit stream being sent to the transmitter. There are ways of estimating BER from the received signal without having to know the transmitted pattern. One way is to look at the eye diagram (see Figure 4.11). As more interference, noise, or other impairments enter the transmission path, the open center of the eye closes. The BER may be estimated from the amount of eye closure.

Another way to estimate BER is to look at the number of times errors are corrected. Most digital systems transmit extra bits for the purpose of correcting errors in the transmission (see Section 3.6.2). When these corrections are made, an error is counted. The bits received are also counted, and the ratio becomes the BER.

BER may be measured before or after error correction. These measurements are sometimes called the raw, or precorrected, BER (measured before error correction) and the corrected, or postcorrection, BER (after error correction). Both measurements have their place. Precorrected BER is the better indication of the channel performance, but postcorrection BER is the better indication of the quality of the signal the user will experience.

Modulation Error Ratio

Figure 4.22(a) plots bit error rate against transmission quality. The transmission quality includes degradation due to noise, distortion, frequency response errors, group delay, echo, gain compression, and other things that can damage the quality of the digital signal. In the left region, where transmission quality is low, the BER specifies the quality of the signal. However, above a certain noise density, the BER is so high we either don’t see errors or at least don’t have the patience to wait long enough to measure them. Even in this region, however, we are interested in knowing the quality of the channel. Since the change in transmission quality between a perfect picture and no picture is very small, it is possible that a technician will leave a home with perfect picture quality, only to have the subscriber call a few minutes later with no picture at all. Thus, it is not enough to know that a picture is good or that data is being transmitted efficiently. We need to know the quality of the channel even when things are going well.

Figure 4.22 (a) BER vs. transmission quality. (b) Definition of MER.

A useful metric for communications quality is the modulation error ratio (MER). MER is defined in Figure 4.22(b), which shows one quadrant of the constellation diagram for a 16-QAM signal. Specialized test equipment is used to display the constellation diagram and to measure the MER. In Figure 4.22(b) the nominal location of each point in the constellation is shown as an open circle. Several pluses mark locations that the constellation point may occupy for some one particular transmitted symbol. In practice, the whole area around the nominal location of the point will be occupied with points, because channel impairments affect each symbol differently. Some common channel impairments are shown in Figures 4.12 and 4.13. MER is a single number used to evaluate the channel quality regardless of the source of impairment.

MER is defined as 10 times the log of the ratio of the RMS error magnitude to the RMS vector magnitude to the nominal position of the point:

The error magnitude and the vector magnitude are defined in Figure 4.22(b). Note that, while the RMS vector magnitude is constant for each point in the constellation, the RMS error magnitude will change for each symbol, since random processes are at work. In order to derive the MER, the test instrument sums the RMS error magnitude over many samples, at each point in the constellation, and averages them.

MER measures the “fuzziness” of the constellation. MER is sometimes described as similar to C/N measurements in analog transmission. While there are similarities (both are measured in decibels and higher numbers are better), there are also differences. Many things other than just poor C/N contribute to degradation of MER. Any transmission channel impairment affects MER, including random noise, group delay, echoes, phase noise, and gain compression.

In order to achieve good performance with 64-QAM, a MER of at least 23 dB is required. For 256-QAM, a MER of at least 28 dB is necessary. A minimum safety margin of 3 dB above these numbers is recommended. Portable instruments available tend to have a maximum BER measurement capability of about 34–38 dB.⁹

A related measurement you may see sometimes is the error vector magnitude, defined as the ratio of RMS error magnitude to the RMS vector magnitude, expressed as a percentage:

The impairments mentioned earlier have different effects on MER and BER. Noise (see Figure 4.17) is the most studied and is well documented. Phase noise and compression affect mostly the outer points of the constellation, so the MER is not affected much, yet BER may be degraded seriously. CW interference affects both MER and BER because all constellation points are impaired. Impulse noise creates havoc with the BER, but because of its transient nature the average impairment is small and MER is still good. Composite second order (CSO) and composite triple beat (CTB), because of their peaks, look more like impulse noise than CW interference. Each impairment has its own “power,” and the MER reflects the sum of these powers.¹⁰

Because digital transmission is not perfect, transmission errors do occur. To render the system operational, channel coding is used to mitigate such transmission errors. A good example is the downstream digital video or cable modem transmission standard, ITU-T J-83 Annex B, which shows three levels of channel coding.

The best-known channel coding is forward error correction (FEC). The data words are grouped in packets, to which are appended Reed-Solomon (R-S) error-correcting words, e.g., 122 data words for six R-S words (7-bit words). At the receiver the R-S codes are used to detect almost all errors and, in addition, can correct one error for each two R-S codes (in this case up to three errors per packet). This is the basis of the pre-FEC and post-FEC bit error rates.

The second channel-coding mechanism is interleaving. Instead of transmitting all the words of each packet in sequence, they are interleaved. That is, adjacent transmitted words comes from different packets. For instance, all the first words of 128 packets, then all the second words, and so on until all the 128th words of all 128 packets are transmitted. At the receiver, the transmitted words are deinter-leaved, to be regrouped in their original packet order. If impulse noise causes errors on many consecutive transmitted words, then without interleaving the R-S could not correct all errors. With interleaving, the grouped errors in transmission are now spread over a larger number of packets, each having only a few correctable errors. Interleaving provides a burst protection period proportional to the length and number of packets interleaved. On the other hand, it introduces a delay, or latency, in transmission and reception due to the storage requirement of the packets. This is why cable modem specifications limit interleave to 128 × 1. Digital video is more tolerant of latency, and deep interleave of 128 × 4 is often used (that is, the interleave is done over 128 × 4 packets, each of 128 words). The protection period is longer, at the expense of larger latency.

The third channel coding present in Annex B is trellis code modulation (TCM). TCM does not allow all combinations of symbol sequences. Rather, it limits the choice of the following symbol to specific ones according to the preceding symbols and specific rules. At reception, the same rules are used to select the most probable word sequence. There is no guarantee that it’s the right one, but it is likely to be. Many errors are corrected, but some are not, activating the second line of defense, the Reed-Solomon FEC. Because it gives the most probable sequence but no decisive error indication, we cannot measure a pre-TCM bit error rate. Finally, Annex A and Annex C do not have TCM and consequently have a worst pre-FEC error rate than Annex B.¹¹